MCTDH wave function

Bonfanti M, Tantardini GF, Hughes KH, Martinazzo R, Burghardt I (2012) Compact MCTDH wave functions for high-dimensional system-bath quantum dynamics. J Phys Chem A 116(46) 11406-11413... 

This shows that p is similar to the well-known one-particle density of electronic structure theory. Diagonalizing the operator p yields the natural populations and the natural orbitals" defined as the eigenvalues and eigenvectors of p(p). Since we are dealing with distinguishable particles we have a separate density matrix for each degree of freedom. The natural populations characterize the contribution of the related natural orbital to the MCTDH wave function. Small natural populations therefore indicate that the MCTDH expansion converges. The natural populations thus provide us with... 

Secondly, we note that the density matrix may become singular. is a Hermitian, positive semidefinite matrix and - as mentioned above - its eigenvalues, called natural weights, characterize the importance of the corresponding natural orbital. A zero eigenvalue occurs if there is a natural orbital (i.e., a linear combination of the single-particle functions) that does not contribute to the MCTDH wave function. Its time evolution may thus be modified by replacing with p( )... 

In closing this section we emphasize that the MCTDH equations of motion conserve the norm of the wave function and, for time-independent Hamiltonians, the mean energy. This follows directly from the variational principle. Moreover, the MCTDH wave function converges toward the numerically exact wave function with increasing numbers of single-particle functions. 

The working equations of these methods are often not compatible with the particular form of the MCTDH wave function, but they usually can be reformulated to take an MCTDH-compatible form (see Ref. 22 for details). 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function fonn Eq. (C.l) could be an exact solution of the Schrddinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

The MCTDH method is to a large extent defined through its ansatz for the wave function" ... 

Fig. 8. The S2 absorption spectrum of the pyrazine molecule (a). Experimental results from Ref. 37. (b) 4-mode model from Ref. 39. (c) 24-mode model from Ref. 11. The model spectra are the Fourier Transform of the autocorrelation function calculated using the MCTDH wave packet propagation method. A damping function of r = 30 fs has been used in (b) and r = 150 fs in (c).

Quantum dynamics simulations of the UV absorption spectrum and of the electronic state population dynamics of the molecule excited by a short laser pulse resonant with the transition to the bright B2u t t ) state, based on the models described in Sect. 5.3 were performed using the MCTDH method in the multi-set formalism (see Sect. 4.2.5 in Chap. 4). For the representation of the Hamiltonian and the wave function, a Hermite polynomial DVR scheme [60] was used for all the degrees of freedom. The number of SPF and primitive basis functions used in the calculations are listed in Table 5.4. Test calculations with both larger primitive and SPF bases have been... 

Let us assume that at r = 0 the wave function If is given in MCTDH form, i.e., given by equation (16). (The question of how to define and generate an initial-state wave function is addressed below.) We want to propagate If while preserving its MCTDH form. As was done above, we derive first-order differential equations (equations of motion) for A and by employing the time-dependent variational principle equation (5). But before doing so, we partition the Hamiltonian H into a separable and residual part ... 

The MCTDH equations of motion are rather complicated, but their use is in general of advantage because there are fewer differential equations to be solved when compared with the standard method of representing the wave function in a time-independent product basis set. In order to investigate under what conditions MCTDH can be expected to be of advantage, we define the effort as the number of multiplications to be carried out. Keeping only the most important terms, one arrives, for a standard method (i.e., using a time-independent DVR), at... 

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